This introduction to the ideas and methods of linear functional analysis shows how familiar and useful concepts from finite-dimensional linear algebra can be extended or generalized to infinite-dimensional spaces. Aimed at advanced undergraduates in mathematics and physics, the book assumes a standard background of linear algebra, real analysis (including the theory of metric spaces), and Lebesgue integration, although an introductory chapter summarizes the requisite material. Highlights of the second edition include a new chapter on the Hahn-Banach theorem and its applications to the theory of duality. This chapter also introduces the basic properties of projection operators on Banach spaces, and weak convergence of sequences in Banach spaces. To begin with, the text develops the theory of infinite-dimensional normed spaces, in particular Hilbert spaces, moving on to study operators between such spaces. The final chapters discuss the particularly important areas of integral and differential equations.

Klappentext

This book provides an introduction to the ideas and methods of linear func­ tional analysis at a level appropriate to the final year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the the­ ory of metric spaces). Part of the development of functional analysis can be traced to attempts to find a suitable framework in which to discuss differential and integral equa­ tions. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions defined on some set. In general, such a vector space is infinite-dimensional. This leads to difficulties in that, although many of the elementary properties of finite-dimensional vector spaces hold in infinite­ dimensional vector spaces, many others do not. For example, in general infinite­ dimensional vector spaces there is no framework in which to make sense of an­ alytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now briefly outline the contents of the book.

Inhalt

1. Preliminaries.- 2. Normed Spaces.- 3. Inner Product Spaces, Hilbert Spaces.- 4. Linear Operators.- 5. Linear Operators on Hilbert Spaces.- 6. Compact Operators.- 7. Integral and Differential Equations.- 8. Solutions to Exercises.- Further Reading.- References.- Notation Index.